#5572271

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1296 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

(2,5)(6,8)(9,10,12,15)(11,13,16,14)(17,18,22,23)(19,20,26,24)(21,25,30,31)(27,28,29,32)(35,41)(39,43)(42,47)(46,48);(9,11)(10,13)(12,16)(14,15)(18,23)(19,26)(25,31)(27,29)(33,34,38,36)(35,39,41,43)(37,44,45,40)(42,48,47,46);(9,11,12,16)(10,14,15,13)(17,19,22,26)(18,24,23,20)(21,29,30,27)(25,28,31,32)(33,35)(34,39)(36,43)(37,42)(38,41)(40,46)(44,48)(45,47);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15)(17,20,22,24)(18,19,23,26)(21,28,30,32)(25,27,31,29)(33,36,38,34)(35,39,41,43)(37,44,45,40)(42,46,47,48);(1,3,4,7)(2,6,5,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=fgh,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5479467);